of 32

ayush-garg
• Category

## Documents

• view

266

2

### Transcript of Unit-1_ Lec 1_ ITA

• 7/24/2019 Unit-1_ Lec 1_ ITA

1/32

Review of Basic Probability

Unit-1

• 7/24/2019 Unit-1_ Lec 1_ ITA

2/32

What is Probability?

Deterministic phenomena

Daily sunrises andsunsets

Tides at sea shores

Phases of the moon

Seasonal changes in

weather

Annual flooding of theYamuna

Random phenomena

Results of coin tosses

Results of rolling dice

Results of horse races

World refer to dice games

and gambling

• 7/24/2019 Unit-1_ Lec 1_ ITA

3/32

Probabilistic notions are common placein everyday language usage

We use words such as

Probable/improbable; possible/impossible

Certain/uncertain; likely/unlikely

Phrases such as there is a 50-50 chance

The probability of precipitation is 20%

• 7/24/2019 Unit-1_ Lec 1_ ITA

4/32

PROBABILITY ?

The theory of probability deals with averagesof mass phenomenon occurring sequentially orsimultaneously.

The purpose of theory is to describe andpredict averages in terms of probabilities ofevents.

• 7/24/2019 Unit-1_ Lec 1_ ITA

5/32

Definition

Three Aproaches to Probability:

Classical

Relative frequency Axiomatic

• 7/24/2019 Unit-1_ Lec 1_ ITA

6/32

Classical (Priori) Approach to Probability:

The probability of Heads is because there are twosides to the coin

This is called the classical approach to probability

More generally, if there are n possible outcomes of anexperiment, then each outcome has probability 1/n

Justification: Symmetry principle (Equally likely);

• 7/24/2019 Unit-1_ Lec 1_ ITA

7/32

Problems with Classical Approach:

What exactly is an outcome?If we toss two coins, are there three outcomes or fouroutcomes?

{(T,T), ( T,H), ( H,T), ( H,H)}?

Note that 2 Heads has probability 1/3 or dependingon the choice

There are only two outcomes: either I Win theLottery, or I dont, so probability is 1/2?

• 7/24/2019 Unit-1_ Lec 1_ ITA

8/32

Relative Frequency (Posteriori) Approach:

the probability of Heads is because when tossed, thecoin will turn up Heads half the time

How do we know the coin will turn up Heads half the

time? Suppose multiple heads tosses have resulted in 50%

Setting P (Head)=1/2 is the relative frequency approach

to probability

• 7/24/2019 Unit-1_ Lec 1_ ITA

9/32

• 7/24/2019 Unit-1_ Lec 1_ ITA

10/32

Relative Frequency (Posteriori)Approach contd

If an outcome x occurs m times on N trails, its

relative frequency is m/N & we define itsprobability P(x) to be m/N

Does there exist a probability of Heads for newunbiased untossed coin?

Or do probabilities come into existence only aftermultiple tosses?

How large N should be?

Are probabilities re-defined after each toss?

• 7/24/2019 Unit-1_ Lec 1_ ITA

11/32

Probability as beliefs:

essentially statements of beliefs A fair coin is one for which P(Heads)=1/2

but how do we know whether a given coin

is fair? Symmetry of the physical object is a belief

That further tosses of a coin for which

• 7/24/2019 Unit-1_ Lec 1_ ITA

12/32

Axiomatic Approach to Probability:

In the axiomatic approach, probabilities are numbers

in the range [0, 1] Certain probabilities are assumed to be given (we

allows the calculations of other probabilities in amathematically & logically consistent manner

It is probability calculas

allows the computation of probabilities withoutrequiring philosophical discussions about the meaningof probability

Consistent with all the approaches described above

• 7/24/2019 Unit-1_ Lec 1_ ITA

13/32

So , t h e P r o b a b i l i t y c a n b e d e f i n e d

a s t h e M e a s u r e o f t h e p o s s i b i l i t i e s o f

o c cu r r e n c e o f a n e v e n t i n a

r a n d o m e x p e r im e n t .

P r o b a b i l i t y o f a n e v e n t A i s

d e n o t e d b y P ( A ) .

• 7/24/2019 Unit-1_ Lec 1_ ITA

14/32

Probability in Engineering:

Thermal noise in electrical circuits

Information Theory

Communication systems design

Noise

Games of Chance

Reliability of systems

Failure probabilities

Failure Rates

Mean time to failure

• 7/24/2019 Unit-1_ Lec 1_ ITA

15/32

Networks & Systems Problems

Random arrivals of packets/jobs

Random lengths/service times

Random requests for resources

Probability of buffer or queue overflow

Transmission or service delays

Scheduling problem, priorities, QOS

Flow control and routing

• 7/24/2019 Unit-1_ Lec 1_ ITA

16/32

Trial ?

Experiement ?

Outcome ?

Sample space ? Event ?

• 7/24/2019 Unit-1_ Lec 1_ ITA

17/32

Experiments and Trials

Fundamental notion: An experiment is

performed and its outcome observed

This is called a trail of experiment

The experiment may be performed by a

human agent, e.g., tossing a coin

or rolling a dice

The experimental outcome might just be

the measurement of a naturally occurringrandom phenomenon, e.g. a noise voltage

• 7/24/2019 Unit-1_ Lec 1_ ITA

18/32

The Sample Space

The set of all possible outcomes of anexperiment is called sample space of the experiment

Examples: The experiment isTossing a coin: ={H,T}

rolling a dice: ={1,2,3,4,5,6}

noise voltage: ={x:-1x1}

• 7/24/2019 Unit-1_ Lec 1_ ITA

19/32

Exercise

Example: The experiment is

rolling a dice: ={1,2,3,4,5,6}

suppose that each outcome is equally likely:

P(1)= P(2)= P(3)= P(4)= P(5)= P(6)

What is probability of rolling an evennumber?

What is probability of rolling an prime

number?

• 7/24/2019 Unit-1_ Lec 1_ ITA

20/32

An even number is said to have beenrolled if the outcome is any of {2,4,6}

P (even number)=1/2; more explicitly

P (even number)=3/6 since 3 of the 6outcomes are in the subset {2,4,6}

An prime number is said to have been

rolled if the outcome is any of {2,3,5} P (prime number)=1/2 also

• 7/24/2019 Unit-1_ Lec 1_ ITA

21/32

Event

A subset of is called an event

Example: A={2,4,6} & B={2,3,5}are said to be events defined onsample space ={1,2,3,4,5,6}

events defined on the sample spaceis merely a probabilists way of sayingsubsets of the sample space

Ac={1,3,5} & Bc={1,4,6} also areevents define on

• 7/24/2019 Unit-1_ Lec 1_ ITA

22/32

When does an event occur?

An event A is said to have occurred on a trial

if the outcome of the trial is a member of thesubset A

Event A occurs if the observed outcome is

some member of A; we dont care whichmember of A it is

If the observed outcome is not a member ofA, then we say A did not occur, orequivalently, we say that Ac occurred

• 7/24/2019 Unit-1_ Lec 1_ ITA

23/32

Outcomes vs. Events

Every trial results in only one outcome, i.e.,

only one of the elements in can be observedoutcome

The observed outcome is a member of several

different subsets, i.e., events & all theseevents are said to have occurred

Fundamental notion: on each trial of the

experiment, one outcomes occurs, butmany events occur

• 7/24/2019 Unit-1_ Lec 1_ ITA

24/32

Example: If outcome of rolling adice is 4, then

Events A ={2,4,6} & Bc={1,4,6} both

have occurred

Events Ac={1,3,5} & B={2,3,5} did

not occur Event A U Bc = {1,2,4,6} has

occurred

Event A and Bc = {4,6} has occurred

• 7/24/2019 Unit-1_ Lec 1_ ITA

25/32

Two special events

can be regarded as a subset of

On any trial, the event always occurs The event is called the certain event or the

sure event

, the empty set, is also a subset of On any trial, the event never occurs

The event is called null event or the impossibleevent

A sample space of n elements has 2n

different subsets including &

• 7/24/2019 Unit-1_ Lec 1_ ITA

26/32

Probabilities of the special events

always occurs; c= never occurs

Conclusion: the probabilities assignedto & should be 1 & 0respectively, regardless of how wechoose to assign probabilities to theoutcomes

P()=1 will be used as an axiom inthe axiomatic approach to probability

• 7/24/2019 Unit-1_ Lec 1_ ITA

27/32

Arbitrary probability assignment

Nonclassical approach: The n outcomes

have probabilities p1, p2 ...pn where pi0& pi =1

The probability of an event A is the sum

of the probabilities of all the outcomesthat comprise A

P(A)=sum of the pi for all members of A

Example: A={x2,x4,x22}P(A)=p2+p4+p22

• 7/24/2019 Unit-1_ Lec 1_ ITA

28/32

Disjoint Events

Events A & B are said to be disjoint or

mutually exclusive if A & B have no element in common

A ={1,3,5} & B={2,4,6} are disjoint events

A U B = {1,2,3,4,5,6}

P(A U B ) =p1+p2+p3+p4+p5+p6= P(A)+P(B)

=I BA

• 7/24/2019 Unit-1_ Lec 1_ ITA

29/32

Probability Axioms for finite spaces

Probabilities are numbers assigned to events

that satisfy the following rules: Axiom I: P(A)0 for all events A

Axiom II: P()=1

Axiom III: If events A & B are disjoint, thenP(A U B ) = P(A)+P(B)

Consequences: P()=0

P(Ac)=1 - P(A); P(A)=1 - P(Ac)

0P(A)1 for all events A

• 7/24/2019 Unit-1_ Lec 1_ ITA

30/32

Countably infinite sample spaces

Let = {x1,x2,.xn,.} be the Countably

infinite sample spaces P{xn}=pn where pn0

For a finite subset A of , P(A) is just the

sum of the probabilities of the outcomescomprising the event A, as before

It seems reasonable to have this idea work

for an infinite subset of A as well But we need a new improved axiom

• 7/24/2019 Unit-1_ Lec 1_ ITA

31/32

New improved Axiom III

Let A1,A2,..An denote a countable sequence

of disjoint events, i.e.,for all ij. Then, P(A1UA2U.. UAnU. )

=P(A1)+ P(A1)+ P(A1)+..P(An)+.

The new axiom implies that P() =0 &

P(A U B ) = P(A)+P(B) for AB =

=I ji AA

• 7/24/2019 Unit-1_ Lec 1_ ITA

32/32

The Probability Space:

Formal statement of the axiomatic

theory A probability space (,F,P) consists of

The sample space consisting of all

possible outcomes of the experiment The -field of events F which includes

all the interesting subsets of

The probability measure P() thatassigns probabilities to the events